1 · The essentials
Start with a polynomial with integer coefficients and evaluate it at positive integers. We look for outputs not divisible by any prime square, such as 4, 9, or 25. For a degree-four polynomial, the manuscript gives the expected proportion of inputs producing such outputs as more and more inputs are counted. It assumes that the polynomial cannot be split into nonconstant factors with integer coefficients and that no single prime square divides all its values. Under these conditions, the resulting proportion is strictly positive. The result extends to degrees four through eight by excluding powers whose exponent is the degree minus two, rather than just squares. The main result has a Lean formalization.
A picture to help (an imperfect image, not to be taken literally)
As an imperfect image, imagine a sequence of tickets, each bearing the polynomial’s value at a positive input. Each prime supplies a filter that rejects tickets whose number is divisible by its square; the result describes the proportion of tickets passing all the filters at once. The image does not explain how infinitely many filters can be controlled.
Why it matters
The result does more than guarantee that squarefree values exist: it gives a positive limiting proportion matching the prediction. For powers with exponent equal to the degree minus two, the results in degrees four through eight combine with Browning’s higher-degree theorem to cover every degree at least four.
2 · Going further
Let be an irreducible polynomial with integer coefficients and degree four. The problem is to determine the density of positive integers for which is squarefree, meaning that no , with prime, divides . The hypothesis excludes a fixed prime-square divisor: no single may divide every polynomial value. The main result gives the positive density predicted by an Euler product, which brings together the divisibility conditions associated with individual primes.
More generally, the manuscript obtains the corresponding density of -power-free values for , under the necessary local condition excluding a fixed divisor of the form . When , this is exactly the squarefree-values statement. The repository summary describes this case as resolving the squarefree-values conjecture for irreducible quartics under the stated hypothesis.
According to the abstract, the argument combines number-field factorization, determinant estimates with adaptively chosen auxiliary primes, and explicit low-degree geometry. The supplied text does not explain how these methods fit together in detail. Only one manuscript is presented; the stated external connection is Browning’s higher-degree theorem, which completes the coverage for all .
Useful notions
- Integer-coefficient polynomials and degree
- Prime numbers and divisibility
- Irreducibility of a polynomial
- Limiting proportions or density
What was known before / what is new
Before
The source places the result in the context of a conjecture about squarefree values of quartic polynomials. It also cites Browning’s theorem for higher degrees, without specifying its threshold or methods here.
New
The manuscript establishes the predicted positive density of squarefree values for irreducible quartics with no fixed prime-square divisor. It also obtains the density of -power-free values in degrees four through eight under the necessary local condition. The main result is formalized in Lean.
Limits and assumptions
The conclusions require irreducibility and the absence of a prime power of the forbidden exponent dividing every value. The manuscript directly treats degrees four through eight; the statement for all degrees at least four also relies on Browning’s theorem. The source reports a Lean formalization of the main result but does not specify its detailed coverage, so formal verification should not automatically be attributed to every mentioned claim. It provides neither a rate of convergence to the density nor a list of remaining open questions.
Mini glossary
- Squarefree value —
- An integer not divisible by the square of any prime.
- Irreducible polynomial —
- A polynomial that cannot be written as a product of two nonconstant polynomials over the coefficient domain under consideration.
- Density —
- The limit of the proportion of inputs satisfying a property among integers from 1 to a bound tending to infinity.
- Euler product —
- A product indexed by primes, here expressing the predicted density through local divisibility conditions.
- -power-free value —
- An integer not divisible by for any prime .
- Fixed prime-square divisor —
- The square of a single prime that divides every value of the polynomial.
Keywords : Squarefree values · Irreducible polynomials · Density
AI-generated explanation based on the manuscript. Unofficial. Refer to the manuscript for the authoritative version.
3 · Technical details and sources
Claim made in the manuscript (source text)
Proves the squarefree-values conjecture for irreducible integer quartics with no fixed prime-square divisor: squarefree values on positive integers have the predicted positive Euler-product density. More generally, establishes the -power-free density for irreducible integer polynomials of degrees four through eight under the necessary local condition; together with Browning's higher-degree theorem, this covers every d ≥ 4.
Verified by Lean: the Lean software checked the part of the reasoning described below, sometimes only part of the manuscript. This does not mean independent mathematicians have validated the result.
Manuscripts in this family
Squarefree values of quartics and power-free values of polynomials
24/09/2026We prove that every irreducible integer quartic with no fixed prime-square divisor takes squarefree values with the predicted positive Euler-product density. More generally, we obtain the corresponding -power-free density in degrees . The proof combines number-field factorization, determinant estimates with adaptive auxiliary primes, and explicit low-degree geometry. Together with Browning's theorem for higher degrees, this gives the -power-free density for every d ≥ 4.